As you can see here, on the left: Piano Keys E flat is the Major Key and C minor is it's accompanying Minor Key.

Yes, the composer chose to write his music in this key and the first four notes are as you want to play them: C, D, D#, C followed by: C, D, D#, C, F# and these are in the keys of E Flat Major and C Minor.

Just some history, if you're interested : The Dragnet Theme was written by Walter Schumann for the radio and television show, but after it's debut, and when it became a hit on the music charts, the 'real' author of the piece, Miklós Rózsa who wrote it for the 1946 film version of "The Killers" challenged the authorship by Shumann. They came to an agreement regarding shared royalties, but Schumann continued to be known as the author of the Dragnet theme. More here: Dragnet Theme

I hope this clarified it for you but, click 'Reply' if you need more help. Thanks!

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